Strictly cyclic algebra of operators acting on Banach spaces H p ( β )

Bahmann Yousefi

Czechoslovak Mathematical Journal (2004)

  • Volume: 54, Issue: 1, page 261-266
  • ISSN: 0011-4642

Abstract

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Let { β ( n ) } n = 0 be a sequence of positive numbers and 1 p < . We consider the space H p ( β ) of all power series f ( z ) = n = 0 f ^ ( n ) z n such that n = 0 | f ^ ( n ) | p β ( n ) p < . We investigate strict cyclicity of H p ( β ) , the weakly closed algebra generated by the operator of multiplication by z acting on H p ( β ) , and determine the maximal ideal space, the dual space and the reflexivity of the algebra H p ( β ) . We also give a necessary condition for a composition operator to be bounded on H p ( β ) when H p ( β ) is strictly cyclic.

How to cite

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Yousefi, Bahmann. "Strictly cyclic algebra of operators acting on Banach spaces $H^p(\beta )$." Czechoslovak Mathematical Journal 54.1 (2004): 261-266. <http://eudml.org/doc/30856>.

@article{Yousefi2004,
abstract = {Let $\lbrace \beta (n)\rbrace ^\{\infty \}_\{n=0\}$ be a sequence of positive numbers and $1 \le p < \infty $. We consider the space $H^\{p\}(\beta )$ of all power series $f(z)=\sum ^\{\infty \}_\{n=0\}\hat\{f\}(n)z^\{n\}$ such that $\sum ^\{\infty \}_\{n=0\}|\hat\{f\}(n)|^\{p\}\beta (n)^\{p\} < \infty $. We investigate strict cyclicity of $H^\{\infty \}_\{p\}(\beta )$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^\{p\}(\beta )$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H^\{\infty \}_\{p\}(\beta )$. We also give a necessary condition for a composition operator to be bounded on $H^\{p\}(\beta )$ when $H^\{\infty \}_\{p\}(\beta )$ is strictly cyclic.},
author = {Yousefi, Bahmann},
journal = {Czechoslovak Mathematical Journal},
keywords = {the Banach space of formal power series associated with a sequence $\beta $; bounded point evaluation; strictly cyclic maximal ideal space; Schatten $p$-class; reflexive algebra; semisimple algebra; composition operator; Banach space of formal power series associated with a sequence; bounded point evaluation; strictly cyclic maximal ideal space; Schatten -class; reflexive algebra; semisimple algebra; composition operator},
language = {eng},
number = {1},
pages = {261-266},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Strictly cyclic algebra of operators acting on Banach spaces $H^p(\beta )$},
url = {http://eudml.org/doc/30856},
volume = {54},
year = {2004},
}

TY - JOUR
AU - Yousefi, Bahmann
TI - Strictly cyclic algebra of operators acting on Banach spaces $H^p(\beta )$
JO - Czechoslovak Mathematical Journal
PY - 2004
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 54
IS - 1
SP - 261
EP - 266
AB - Let $\lbrace \beta (n)\rbrace ^{\infty }_{n=0}$ be a sequence of positive numbers and $1 \le p < \infty $. We consider the space $H^{p}(\beta )$ of all power series $f(z)=\sum ^{\infty }_{n=0}\hat{f}(n)z^{n}$ such that $\sum ^{\infty }_{n=0}|\hat{f}(n)|^{p}\beta (n)^{p} < \infty $. We investigate strict cyclicity of $H^{\infty }_{p}(\beta )$, the weakly closed algebra generated by the operator of multiplication by $z$ acting on $H^{p}(\beta )$, and determine the maximal ideal space, the dual space and the reflexivity of the algebra $H^{\infty }_{p}(\beta )$. We also give a necessary condition for a composition operator to be bounded on $H^{p}(\beta )$ when $H^{\infty }_{p}(\beta )$ is strictly cyclic.
LA - eng
KW - the Banach space of formal power series associated with a sequence $\beta $; bounded point evaluation; strictly cyclic maximal ideal space; Schatten $p$-class; reflexive algebra; semisimple algebra; composition operator; Banach space of formal power series associated with a sequence; bounded point evaluation; strictly cyclic maximal ideal space; Schatten -class; reflexive algebra; semisimple algebra; composition operator
UR - http://eudml.org/doc/30856
ER -

References

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  1. A Course in Functional Analysis, Springer-Verlag, New York, 1985. (1985) Zbl0558.46001MR0768926
  2. The Theory of Subnormal Operators, American Mathematical Society, 1991. (1991) Zbl0743.47012MR1112128
  3. 10.1155/S0161171295000147, Internat. J.  Math. Sci. 18 (1995), 107–110. (1995) MR1311579DOI10.1155/S0161171295000147
  4. Weighted shift operators and analytic function theory, Math. Survey, A.M.S. Providence 13 (1974), 49–128. (1974) Zbl0303.47021MR0361899
  5. On the space  p ( β ) , Rend. Circ. Mat. Palermo Serie  II XLIX (2000), 115–120. (2000) MR1753456
  6. Bounded analytic structure of the Banach space of formal power series, Rend. Circ. Mat. Palermo Serie II LI (2002), 403–410. (2002) MR1947463

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